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You are wise to recognize that no matter how we receive income we are responsible to pay income, social security & medicare tax.
Here is a link to IRS Publication 17. You will find, on page 7, information which will be helpful to you. http://www.irs.gov/pub/irs-pdf/p17.pdf
More specifically if you are being paid "under the table" (which is an illegal act by the employer), or receive a 1099 you will need to report your income on IRS Form 1040 using Schedule C & Form SE (link to forms & instructions attached):
The reason you must use a Schedule C is, unless you receive a W-2 from your employer then, in your case, are will be classified as "self-employed".
You should also, by the end of each 1/4, file and pay an estimated tax. Here is a link to help explain that process:
I am going to provide links to help you determine how much you should set aside to pay your taxes.
In addition to the IRS, New York State & New York City each require you pay income tax as well.
Here is a link to the New York City Tax Tables:
Here is a table for New York State Taxes:
For single taxpayers:
-- 4 percent on the first $8,000 of taxable income.
-- 4.5 percent on taxable income between $8,001 and $11,000.
-- 5.25 percent on taxable income between $11,001 and $13,000.
-- 5.9 percent on taxable income between $13,001 and $20,000.
-- 6.85 percent on taxable income of $20,001 and above.
You can get tax forms for New York State at: http://www.tax.state.ny.us/forms/default.htm
I hope this information is helpful to you.
If you have any further questions on this topic, be sure and get back to me.